**5. Conclusions**

High-order reconstructed discontinuous Galerkin methods based on first-order hyperbolic systems for Navier–Stokes equations have been developed and presented. The presented work is based on a new formulation for the hyperbolic Navier–Stokes system, namely HNS20G, which introduces the gradients of density, velocity, and temperature as additional variables. The numerical cases demonstrate that the presented method is compact, efficient, and robust.

#### **Figure 10.**

*Zoom-in meshes for von Kármán vortex street.*

**Figure 11.**

*Vorticity magnitude for von Kármán vortex street behind a cylinder at Re* <sup>∞</sup> ¼ 100 *within half cycle.*
